Circular dichroism in molecular-frame photoelectron angular distributions in the dissociative photoionization of H
2and D
2molecules
J. F. P´erez-Torres,1,*J. L. Sanz-Vicario,2,†K. Veyrinas,3P. Billaud,3,‡Y. J. Picard,3C. Elkharrat,3S. Marggi Poullain,3,§
N. Saquet,3,M. Lebech,4J. C. Houver,3F. Mart´ın,1,5and D. Dowek3
1Departamento de Qu´ımica, M´odulo 13, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain
2Grupo de F´ısica At´omica y Molecular, Instituto de F´ısica, Universidad de Antioquia, Medell´ın, Colombia
3Institut des Sciences Mol´eculaires d’Orsay, Universit´e Paris-Sud et CNRS, Batiment 210-350, Universit´e Paris-Sud, F-91405 Orsay Cedex, France
4Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
5Instituto Madrile˜no de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain (Received 25 July 2014; published 28 October 2014)
The presence of net circular dichroism in the photoionization of nonchiral homonuclear molecules has been put in evidence recently through the measurement of molecular-frame photoelectron angular distributions in dissociative photoionization of H2[Dowek et al.,Phys. Rev. Lett. 104,233003(2010)]. In this work we present a detailed study of circular dichroism in the photoelectron angular distributions of H2and D2molecules, oriented perpendicularly to the propagation vector of the circularly polarized light, at different photon energies (20, 27, and 32.5 eV). Circular dichroism in the angular distributions at 20 and to a large extent 27 eV exhibits the usual pattern in which inversion symmetry is preserved. In contrast, at 32.5 eV, the inversion symmetry breaks down, which eventually leads to total circular dichroism after integration over the polar emission angle. Time-dependent ab initio calculations support and explain the observed results for H2in terms of quantum interferences between direct photoionization and delayed autoionization from theQ1 andQ2 doubly excited states into ionic states (1sσgand 2pσu) of different inversion symmetry. Nevertheless, for D2at 32.5 eV, there is a particular case where theory and experiment disagree in the magnitude of the symmetry breaking: when D+ions are produced with an energy of around 5 eV. This reflects the subleties associated to such simple molecules when exposed to this fine scrutiny.
DOI:10.1103/PhysRevA.90.043417 PACS number(s): 33.80.Eh, 42.65.Re I. INTRODUCTION
Photoelectron angular distributions of molecules are use- ful to obtain information about the electronic dynamics in photoionization, including the interplay between electronic and nuclear motions. Those arising from randomly oriented diatomic molecules in gas phase, depending only on two spher- ical harmonics, were the subject of early theoretical investi- gations [1,2]. For molecules with a well-defined orientation, angular distributions are much more complex [2] but provide very detailed information on the dynamics of processes such as dissociative photoionization (DPI). Experiments in the gas phase in which the orientation of the molecule at the time of photoejection is fully determined are feasible in current laboratories, thanks to multicoincidence detection techniques such as reaction microscopes [3] or vector correlation methods [4,5]. When these techniques are applied to DPI of diatomic
*Present address: Institut f¨ur Chemie und Biochemie, Freie Univer- sit¨at Berlin, Takustrasse 3, 14195 Berlin, Germany.
†Corresponding author; [email protected]; Present ad- dress: Departamento de Qu´ımica, M´odulo 13, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain.
‡Present address: Laboratoire Aim´e Cotton, CNRS, Batiment 505, F-91405 Orsay Cedex, France.
§Present address: Departamento de Qu´ımica F´ısica, Facultad de Ciencias Qu´ımicas (Unidad asociada CSIC), Universidad Com- plutense de Madrid, 28040 Madrid, Spain.
Present address: School of Chemistry, University of Nottingham, Nottingham NG7 2RD, United Kingdom.
molecules [AB+ ω(ˆp)→ A++ B + e−], the triplet of vectors (VA+,Ve,ˆp) corresponding to the velocity of the ionic fragment, the velocity of the ejected electron, and the field polarization axis, respectively, can be measured in coincidence for each DPI event. Therefore, they allow one to obtain a complete kinematical description of the ionization dynamics for oriented or fixed-in-space molecules in the gas phase.
Circular dichroism (CD) represents the different response of a system when exposed to left and right circularly polarized light. CD in photoabsorption of chiral molecules, i.e., without a plane or center of symmetry, is an effect of optical activity already known since its discovery by Pasteur [6]. For such species, the study of photoelectron circular dichroism (PECD) has encountered numerous developments in recent years and has been demonstrated to be a direct and orbital-sensitive chiroptical probe of static and dynamical molecular structures (see [7] and references therein). Recent applications address the determination of absolute molecular stereochemistry in the gas phase [8]. Nonchiral molecules, like diatomic molecules, may also show particular asymmetries with respect to different photon helicities. In spite of the lack of molecular chirality, a handedness may be induced in the photon-molecule system by the experimental geometry, considering enantiomorphic arrangements of the vectors corresponding to molecular orientation, photon propagation and electron ejection [9].
The circular dichroism in photoelectron angular distri- butions (CDAD) corresponds to the difference in the pho- toionization cross sections, differential in the emission polar angle θerelative to the molecular axis [see Fig.1(a)], for left and right circularly polarized light. The CDAD of molecular
, (1sσ )
H2, X1Σ+g
0 1 2 3 4 5 6
Internuclear distance [a.u.]
0 5 10 15 20 25 30 35 40
Energy [eV]
2Σ+(2pσu) Q1
Q2
2Σ+g(1sσg)
(b)
(a)
FIG. 1. (Color online) (a) Photoionization axis frame. The molecule is oriented along the Z axis. The light propagation direction is represented by the vector k along the X axis. The electron is ionized in the direction given by e= (θe,φe) and has a momentum vector ke. The proton is ejected in the upward direction (χ= 90◦) along the molecular axis n. All results shown in the manuscript correspond to the Y Z plane (i.e., χ= 90◦, φe= 90◦, or φe= 270◦). (b) Potential energy curves of H2 and H2+. X1g+denotes the ground electronic state of H2and2g+(1sσg) and2u+(2pσu), the ground and the first excited electronic states of H2+(i.e., the first and the second ionization thresholds of H2, respectively). A photon of 32.5 eV (orange vertical line) is absorbed by the H2molecule within the Franck-Condon region enclosed by dashed lines. The decay of the two lowest1+u 1Q1and
1 u1Q2autoionizing states is indicated qualitatively with arrows.
photoelectrons from nonchiral oriented molecules was first investigated theoretically by Ritchie [10], who pointed out that such particular effects may appear due to interferences between the dipolar and higher-order terms in the multipole expansion of the radiation field. Cherepkov [11] predicted the existence of circular dichroism in the angular distributions of photoelectrons from oriented chiral molecules within the dipole approximation but assuming spin-orbit interaction. In a more detailed analysis [12,13], it was demonstrated that CDAD can also exist in oriented linear molecules even within the electric dipole approximation. Three major features were pointed out in the latter case: (i) CDAD in electron pho- toemission arises from the interference between degenerate continuum partial waves with m values differing by ±1.
(ii) CDAD vanishes if the triplet of vectors (n,ke,k) corre- sponding to the orientation of the molecular axis, the direction of the photoelectron momentum, and the direction of propaga- tion of the circularly polarized light, respectively, are coplanar.
(iii) For linear molecules with an inversion center, CDAD does not exist in the reflection plane perpendicular to the molecular axis. For other geometrical arrangements, such as the one adopted in the present work [see Fig. 1(a)] for dissociative ionization of H2and D2, CDAD should be present.
Motivated by these predictions, Westphal et al. [14,15]
reported the first experimental evidence of CDAD of pho- toelectrons ejected from oriented CO molecules, adsorbed on a crystal metal surface of Pd(111). Reid et al. [16] produced a first experiment in the gas phase with a complete description of the molecular photoionization, by measuring rotationally resolved CDAD with excited NO molecules. Completeness refers here to the determination of all complex dipolar transition matrix elements and their phases for each partial wave in the continuum. This kind of complete experiments has been extended nowadays to photoionization of molecules
in the ground state, taking advantage of dissociative ionization and using multicoincidence detection methods like those mentioned above. By using these methods, CDAD has been observed [17–25] in molecular-frame photoelectron angular distributions (MFPADs) of oriented achiral molecules such as N2, CO, NO, O2, N2O, H2, and D2, following direct or resonant inner- and valence-shell photoionization.
Multicoincidence detection methods are able to elucidate in which direction along the molecular axis the photoion is ejected. Thus, the experiment itself imposes boundary conditions for the ion (electron) localization after the prompt dissociation following the photoionization. This implies that the final total wave function is effectively projected onto stationary continuum states that localize one of the ions (or the remaining bound electron) in a given center. Assuming the geometry shown in Fig. 1, photoions may be released upwards or downwards, so that the delocalized nature of the remaining bound electron in H2+ (D2+) (a signature of the inversion symmetry) turns out to be now localized [H(n)+ H+or H++H(n)], i.e., the localized wave functions become ϕ1s =√12(φ1σg± φ1σu). A discussion on this issue can be found in [26,27], in which MFPADs in dissociative photoionization of H2and D2 are comprehensively analyzed for the case of linearly polarized light by using a time- independent perturbation theory.
One-photon absorption with circularly polarized light pho- toionizes H2(D2) molecules from the ground state X1g+to final continuum states1u+and1 u. For the photon energies considered in this work, only H2+ionic thresholds2g+(1sσg) and2+u(2pσu) need to be considered [see Fig. 1(b)]. This implies that partial waves of the escaping photoelectron may have angular momentum odd (ungerade continuum orbitals for 1σgεchannels) and even (gerade continuum orbitals for 1σuε channels) in both1u+ and1 u final symmetries.
As we will see below, the origin of the observed CDAD and the presence of net circular dichroism CD after integration on the polar angle θe (even-odd) interference in the final-state partial waves with opposite inversion symmetry and m values differing by±1.
From our previous studies in H2 [28] we have learned that CD effects occur at proton kinetic energies where DPI through both the 1sσgand 2pσuchannels is possible. At these proton kinetic energies, CD is almost entirely determined by autoionization from theQ1 andQ2 doubly excited states of H2. Although the autoionization lifetimes in H2 and D2 are identical, one can expect that CD is different, because, due to the different mass of the nuclei, the Franck Condon region in D2 is narrower than in H2, which implies differences in the photoexcitation process, and nuclear motion is slower, which modifies the dissociation dynamics.
In this work, we have theoretically and experimentally investigated CDAD of H2 and D2 and we report results in terms of photoelectron emission in the plane that contains the molecular axis and is perpendicular to the light propagation direction k for three photon energies: 20, 27, and 32.5 eV. This is an extension of our previous work reported in [28], where a more limited photon energy range was considered and resonant effects were investigated only for H2. The present results show, in agreement with what was anticipated in Ref. [28]
for H2, that for both molecular isotopes CD is possible due to the interference between direct photoionization and delayed autoionization from theQ1andQ2doubly excited states into ionic H2+ states of different inversion symmetry (1sσg and 2pσu). These interferences change dramatically as a function of the nuclear kinetic energy [29]. The present experimental results show the same qualitative behavior for H2and D2at the three photon energies, while the theory predicts that this be- havior should be less similar at 32.5 eV (where autoionizations fromQ1andQ2resonances are simultaneously relevant). The paper is organized as follows. In Sec.II, we introduce the basic formalism for CDAD and in Sec.IIIour time-dependent theory to obtain CDAD. Experimental and computational details are given in Secs.VandIV, respectively. In Sec.VI, we present our results and discussion. Atomic units are used throughout unless otherwise stated.
II. CIRCULAR DICHROISM FOR PHOTOEMISSION IN THE MOLECULAR FRAME
The molecular-frame photoelectron angular distribution I(χ ,θe,φe) for the geometry shown in Fig.1takes a remarkably simple general form in the dipole approximation for incident circularly polarized light [20,30] [here μ0= 0 is for linearly polarized light and, for instance, μ0 = +1 indicates left- handed circularly (LHC) polarized light of positive helicity h= +1]:
Iμ0=±1,χ,φe(θe)= F00(θe)−12F20(θe)P20(cos χ )
−12F21(θe)P21(cos χ ) cos(φe)
−12F22(θe)P22(cos χ ) cos(2φe)
± F11(θe)P11(cos χ ) sin(φe), (1) where χ is the polar angle (0 χ π) indicating the orientation of the molecular axis n with respect to the light propagation axis k and the set (θe,φe) indicates the direction of the electron emission vector ke in the molecular frame, defined by the molecular axis and the light propagation axis.
In Eq. (1), the dependence on the azimuthal angle φe
and the molecular orientation angle χ is factorized in terms of simple trigonometric functions and associated Legendre polynomials PLN, respectively, so that complete dynamical information about the dissociative photoionization reaction is fully described by the five FLN(θe) functions [31]. These functions can be determined experimentally by performing a Legendre-Fourier analysis in (χ , φe) of the MFPAD I(χ ,θe,φe), using expression (1). The FLN functions depend on the polar angle θe and the total energy (electronic and nuclear) of the final continuum state. To emphasize the geometrical aspect of these functions, the latter dependence has been omitted in the notation. When photoionization is fast in comparison with dissociation of the molecular ion [20], variations with total energy are entirely due to variations in the photoelectron energy, since the nuclei barely move and therefore their kinetic energy remains nearly constant [20].
However, when, as in the present study, both ionization and dissociation occur on a comparable time scale, the energy in the final state is shared between electrons and nuclei, and consequently the FLN(θe) functions do vary with the photoion
kinetic energy [28,32]. We will use this dependence upon θe
and the photoion kinetic energy to present and discuss our results. In particular, the F21 function gives access to the absolute value of the relative phase between the amplitudes for parallel (1g+→1+u) and perpendicular (1g+→1 u) dipole transitions, whereas F11 (accessed only by using circularly polarized light) provides the sign of this relative phase and determines the presence of circular dichroism. Once these five functions are extracted experimentally or calculated, MFPADs for any orientation χ of the molecular axis can be fully reconstructed.
The observation of CDAD in the molecular frame is largest for electron scattering in a plane Y Z perpendicular to the light propagation axis (φe= 90◦or 270◦) and containing the molecular axis (χ = 90◦), due to the sin φe dependence in Eq. (1). For the geometry adopted in this work, χ= 90◦, Eq. (1) reduces to
Iμ0=±1,χ=90,φe(θe)= F00(θe)+14F20(θe)−32F22(θe) cos(2φe)
± F11(θe) sin(φe), (2) and only four FLN functions are needed. In this Y Z plane, CDAD is characterized as the relative variation of the MFPAD I(χ = 90◦,θe,φe= 90◦) [or I (χ= 90◦,θe,φe= 270◦)] when the helicity of the light is changed from h= +1 (LHC polarized) to h= −1 (RHC polarized). In fact, CDAD can be defined in two equivalent forms, and expressed straightfor- wardly in terms of the FLN functions:
CDADχ=90,φe=90(θe)=I+1,90,90− I−1,90,90 I+1,90,90+ I−1,90,90
= CDADχ=90,h=+1(θe)= I+1,90,90− I+1,90,270
I+1,90,90+ I+1,90,270
= 2F11
2F00+12F20+ 3F22
. (3)
Accordingly, CDAD is driven by the function F11and takes values in the interval [−1,+1]. In most experiments performed so far on homonuclear diatomic molecules [18,22–24], CDAD exhibits an antisymmetric behavior with respect to the polar electron emission angle θe in the plane perpendicular to the light propagation. In this case, the net circular dichroism CD, defined by integrating over the polar angle, i.e.,
CD=
sin θedθe(I+1,90,90− I−1,90,90)
sin θedθe(I+1,90,90+ I−1,90,90) (4) vanishes identically. However, we have recently found [28]
that CDAD in resonant DPI of H2(for impact photon energies in the range 30–35 eV) may strongly depart from the expected antisymmetry in θe in specific regions of the ion kinetic- energy release spectrum, and thus the θe-integrated CD does not vanish, but shows instead a richer structure attributed to delayed autoionization of Qn molecular resonances into channels with different inversion symmetry.
III. THEORETICAL METHOD
MFPADs obtained through DPI of simple molecules can be evaluated by using time-independent theories. For instance, the multichannel Schwinger configuration-interaction method
(MC-SCI) of Lucchese and co-workers [33–35] has been shown to provide accurate predictions of molecular-frame photoemission observables measured for a series of molec- ular targets [20–22,30,31]. Density functional theory (DFT) calculations have also been conducted and predict richly structured measured angular distributions from fixed-in-space small polyatomic molecules [36], with recent developments involving time-dependent DFT [37]. All these calculations were performed within the fixed-nuclei approximation. How- ever, the study of resonant DPI, when autoionization and dissociation occur on a comparable time scale, requires a more sophisticated theory since the nuclei have time to move before the electron is ejected. This can be done only by using a full quantum-mechanical treatment of both the electronic and nuclear motions [38]. This time-independent theory along with Dill’s formulas [2] have been used to compute MFPADs in H2
and D2for linearly polarized light [27] as well as for circularly polarized light [23,39]. Here we use instead a recent extension of this methodology to the time domain [40], which allows for the temporal scrutiny of the resonant photodynamics involved in CDAD. The adiabatic and the dipolar approximations are assumed. The axial recoil approximation [41] is also invoked since superexcited states in H2 dissociate faster than the rotational period of the molecule. In practice, this means that the set (VH+,Ve,ˆp) in the vector correlation method are connected with the triplet of vectors (n,ke,k), and corrections due to rotational motion are neglected.
Our method [40] is based on the solution of the time- dependent Schr¨odinger equation (TDSE):
[H0(r,R)+ Vμ0(t)− i∂t](r,R,t)= 0, (5) where r labels the electronic coordinates r1 and r2, R is the internuclear distance, andH0is the H2field-free nonrelativistic Hamiltonian neglecting the mass polarization term
H0(r,R)= T (R) + Hel(r,R). (6) Here T (R)= −∇R2/2μ,Helis the electronic Hamiltonian, and Vμ0 is the laser-molecule interaction potential in the velocity gauge
Vμ0(t)= (p1+ p2)· Aμ0(t), (7) where pi is the linear momentum operator for the electron i and Aμ0 is the vector potential of the radiation field with polarization μ0. The dipolar operator is commonly referred to the laboratory frame but one can make use of the Wigner rotation matrices to express the operator in the molecular frame [2]. Within the dipole approximation, for circularly polarized light defined by the parameter μ0and considering the geometry of Fig.1, we choose the vector potential to be different from zero in the time interval [0,T ] according to the formula
Aμ0(t)= A0sin2
π Tt
⎛⎝
0 μ0cos
ω t−T2 sin
ω t−T2
⎞
⎠, (8)
where A0 is the vector potential amplitude, which is related to the laser peak intensity by A0/ω= [I (W/cm2)/3.5095× 1016]1/2.
The time-dependent wave function (r,R,t) is expanded in a basis of fully correlated adiabatic Born-Oppenheimer
(BO) vibronic stationary states of energy Wk, which include the bound states, the resonant doubly excited states, and the nonresonant continuum states of H2[40]:
(r,R,t)
= Cgνg(t)φg(r,R)χνg(R)e−iWgνgt
+
r
νr
Crνr(t)φr(r,R)χνr(R)e−iWrνrt
+
αm
dε
να
Cανεm
α(t)ψαεm(r,R)χνα(R)e−iWαναt, (9) where φg, φr, and ψαεmrepresent the ground, doubly excited, and continuum electronic states of H2 (or D2), respectively.
Here, α represents the full set of quantum numbers for the electronic state of the residual molecular ion H2+ (or D2+) with BO energy Eα(R) and the indices ε, , and m correspond, respectively, to the kinetic energy, angular momentum, and the Zcomponent of the angular momentum of the ejected electron.
The vibronic states in Eq. (9) result from the solution of the following eigenvalue equations [40]: (i) for the electronic motion
[Hel− Eg(R)]φg = 0, (10) [QHelQ − Er(R)]φr = 0, (11) [PHelP − Eα(R)]ψαεm= 0, (12) where P and Q = 1 − P are Feshbach projection operators that project onto the nonresonant and resonant parts of the continuum wave function, respectively, and (ii) for the nuclear motion
[T (R)+ Eg(R)− Wg,νg]χνg= 0, (13) [T (R)+ Er(R)− Wr,νr]χνr = 0, (14) [T (R)+ Eα(R)+ ε − Wα,να]χνα = 0, (15) where Wxνxrefers to the total vibronic energy.
The electronic wave functions φr and ψαεmare not eigen- functions of the electronic Hamiltonian, which can be written as a sum of projected operatorsHel= QHelQ + PHelP + QHelP + PHelQ. The QHelP term describes the interaction between the Q and P subspaces, which is responsible for the autoionizing decay of the doubly excited states into the continuum. Thus, by introducing the ansatz (9) into the TDSE (5) and projecting onto the basis of stationary vibronic states, one arrives at a set of coupled linear differential equations in which these couplings, in addition to the dipole-induced ones, explicitly appear [40],
id dt
⎛
⎜⎝ Cgνg
Crνr Cανεm
α
⎞
⎟⎠
=
⎛
⎜⎝
0 A(t)Vgνrνgr A(t)Vgνανgαεm A(t)Vrνgνrg 0 QHPανrνrαεm
A(t)Vανgναgεm PHQrνανrαεm 0
⎞
⎟⎠
⎛
⎜⎝ Cgνg Crνr
Cανεm
α
⎞
⎟⎠,
(16)
where each matrix term represents a matrix block and we have neglected nonadiabatic electrostatic couplings. This system of equations must be integrated for t T . This time must be larger than the lifetime of theQ1,2resonant states to allow for their complete decay [40]. In the velocity gauge, the operator pj μ0= −i∇j μ0 is −i∂/∂zj for μ0 = 0 (linear polarization along the molecular axis) and−i(∓∂/∂zj ± i∂/∂yj)/√
2 for μ0= ±1 (circular polarization with components in the Y Z plane). Given that the light propagation vector lies on the X axis (see Fig. 1) one needs to consider only the Y and Z components of the dipolar operator. In this work we will consider only very weak intensities, so that the population of the initial state X1g+remains very close to 1 for all times and multiphoton absorption is negligible compared to one-photon absorption. Thus, in practice, we can consider two sets of coupled equations, the first one for final states1u+ (parallel transitions to states with = 0 and m = 0) and the second one for final states 1 u (perpendicular transitions to states with
= 1 and m = ±1). After propagating both sets separately, the asymptotic amplitudes are collected and then added coherently in the differential cross sections as explained below.
The asymptotic wave function that describes an ejected electron in channel α satisfying incoming wave boundary conditions admits a partial-wave decomposition [2]:
α(−)(r,R,t)=
m
ie−iσ(ε)Ym∗(θe,φe)
× ψαεm(r,R)χνα(R)e−iWαναt, (17)
where σ(ε)= arg( + 1 − i/√
2ε) is the Coulomb phase shift. As mentioned in the Introduction, vector correlation methods are capable of detecting the momentum and direction of all ejected charged particles, and specifically of elucidating in which direction along the internuclear distance axis the proton escaped. With reference to Fig. 1, protons may be released upwards (U ) or downwards (D) and, consequently, a combination of the partial-wave expansions for the 1sσg
and 2pσu channels must be performed to fulfill the specific asymptotic condition [26]
U,D(−) = 1
√2
1sσ(−)
g± 2pσ(−)u
, (18)
which implies that final transition amplitudes (C1sσεmgν,C2pσεm
uν) are added for the U case and substracted for the D case.
Molecular-frame photoelectron angular distributions or, equivalently, photoionization probabilities differential in pro- ton energy (EH+= Wα,να − W∞, where W∞ is the energy of a H atom infinitely separated from H+) and in the solid angle of the ionized electron, can be evaluated fol- lowing Dill’s procedure [2], for any arbitrary polarization μ0, replacing the dipolar transition matrix elements by the asymptotic transition amplitudes obtained after integrating the TDSE:1
d3Pμ0 dndEH+de =
dε
αaαb
μaμb
ama
bmb
i(a−b)ei[σb(ε)−σa(ε)](−1)mb+μa−μ0
Cαεaaνmaμa ∗
Cαεbbνmbμb
×
Le
(2a+ 1)(2b+ 1)
2Le+ 1 ab00|Le0 ab− mamb|LeMe YLMee∗(θe,φe)
×
L
1
2L+ 1 11 − μaμb|LN 11 − μ0μ0|L0 YLN(θn,φn), (19)
with
μa,b= 0, ±1, N = −μa+ μb, Me= −ma+ mb, a+ b Le |a− b| and 0 L 2.
In this equation, e= (θe,φe) is the solid angle for the pho- toelectron emission direction, χ≡ n= (θn,φn) (see Fig.1), j1j2m1m2|J M denotes a Clebsch-Gordan coefficient, and Cαεa,ba,bνmαa,ba,bμa,bis the transition amplitude to a continuum vibronic state for a given value of μa,b and for the given channel
1Note that, in Eq. (1) of Ref. [27], a factor 4π2ω/cwas introduced, so that the square of the transition amplitude directly leads to cross sections in the framework of time-independent perturbation theory.
In the present work, approximate cross sections may be retrieved by multiplying the asymptotic transition probabilities by a factor 3ω/8T I [40].
αa,b= U or D as defined in Eq. (18). Note that in Eq. (19) the first summation over αa and αb runs over the two ionization channels corresponding to the 1sσgand 2pσustates, for both the parallel (1g →1u) and the perpendicular (1g→1 u) transitions. The indices a and b represent the angular momenta of the ejected electron for each channel and final symmetry.
As shown in [31], Eq. (19) for MFPADs is formally and computationally equivalent to Eq. (1), expressed in terms of FLNfunctions and simple trigonometric functions. To see the connection, the FLN functions can be partial-wave expanded in terms of Legendre polynomials and transition amplitudes as a function of the total vibronic energy Wανα, or equivalently, the proton kinetic energy EH+:
FLN(EH+,θe)=
a+b
Le=|a−b|
DLμ0
eLN(EH+)PLNe(cos θe), (20)
where DLμ0
eLN(EH+)
= (−1)N(−i)δL,1δN,1 1+ δN,0
i(a−b)
×
αaαb
μaμb
ama
bmb
(−1)mb+μa−μ0
×
(2a+ 1)(2b+ 1)(Le− N)!(L − N)!
(Le+ N)!(L + N)!
× ab00|Le0 abma− mb|Le− N
× 11 − μaμb|LN 11 − μ0μ0|L0
×
dε
Cαεamaμa
aν
∗ Cαεbmbμb
bν
ei[σb(ε)−σa(ε)]. (21)
The FLN functions can be accessed experimentally as a function of the proton kinetic energy, so that a direct com- parison between theory and experimental results for MFPADs is already feasible at this partial-wave level, providing the strongest test of theory and experiment, since it is done at the level of transition amplitudes and their relative partial-wave phase shifts.
IV. COMPUTATIONAL DETAILS
A detailed description of this method can be found in [38,40,42]. Briefly, a standard configuration-interaction (CI) method is used to obtain the H2 ground state X1g+ and bound excited state, whereas a truncated CI method compatible with the Feshbach formalism is used to obtain the resonant states Q1,21u+ and Q1,21 u. The CI method uses H+2 orbitals expanded in one center, in terms of a basis set consisting of 180 B splines of order k= 8, including angular momenta from = 0 up to = 16 enclosed in an electronic box of size 60 a.u. For a more detailed description of B splines and their applications in atomic and molecular physics, see [42]. Nonresonant electronic continuum states ψαεm are evaluated by using anL2 close-coupling approach.
The uncoupled continuum states (UCSs) that enter in this formalism are built up as antisymmetrized configurations of the type [nλπ(r1),nλ()π (r2)], where nλπ corresponds to a H2+target state and nλ()π is a H2+orbital that represents the ionizing electron within the subspace of angular momentum and parity π . For instance, the configurations for the four continua used in this work have the following forms:
(a) two continua of symmetry1u+built up with configura- tions 1sσgnσu()(n= 1,75) for channels with = 1, 3, 5, and 7, and 2pσunσg()(n= 1,75) for = 0, 2, 4, and 6;
(b) two continua of symmetry 1 u built up with config- urations 1sσgnπu() (n= 1,75) for = 1, 3, 5, and 7, and 2pσunπg()(n= 1,75) for = 2, 4, 6, and 8.
These UCSs are then energy normalized and interchan- nel coupling between the different degenerate channels is introduced by solving the Lippman-Schwinger equation [38,42,43]. The vibrational wave functions χνg, χνr, and χνα
are also expanded in terms of a B-spline basis set, using 240 B splines of order k= 8 inside a box of Rmax= 12 a.u. For the time-dependent solution, we use a sixth-order
Runge-Kutta integrator to solve a system of equations that reaches dimensions up to 30 000 vibronic states. Experimen- tal results on circular dichroism have been obtained with synchrotron radiation. This condition may be achieved in a time-dependent methodology by propagating the TDSE using laser pulses with relatively low intensities and long duration. Such perturbative stationary conditions are met with a pulse duration T = 10 fs and a laser peak intensity of I = 1012W/cm2. Transition amplitudes Cανεmα (t > T ) (which also carry the short-range phase shift) for1u+and1 ufinal states are collected for all partial waves in a discretized set of continuum energies ε and then inserted in Eq. (19) or (21).
V. EXPERIMENTAL DETAILS
A description of the experimental setup used in applying the vector correlation (VC) method for the most recent measurements of MFPADs in D2has already been outlined in [44]. It allows measuring in coincidence the ejection velocity vectors of photoelectrons and photoions emitted in the same DPI event for molecules in the gas phase subject to synchrotron radiation. A double electron-ion velocity spectrometer [5]
combines time-of-flight- (TOF-) resolved ion-electron co- incidence detection and imaging techniques with position- sensitive detection. A collimated beam of H2(D2) molecules emerges from the supersonic molecular jet SAPHIRS [45] and the circularly polarized light is provided by the VUV beamline DESIRS at the third-generation synchrotron radiation facility SOLEIL, operated in the eight-bunch mode with a period T = 147 ns and a pulse width limited to t = 50 ps. The circular polarization rate s3/s0of the light, where (s0,s1,s2,s3) are the Stokes parameters, was higher than 95%. The two beams intersect at right angles in the interaction region located at the center of the spectrometer.
Electrons and ions resulting from DPI are extracted from the interaction region by a dc uniform electric field (from 15 V/cm for photons of ω = 19 eV up to 150 V/cm for photon energies of ω = 32.5 eV), then focused with two electrostatic lenses ensuring a 4π collection of both particles in the energy region of interest, and finally collected by the two-delay line position-sensitive detectors (PSDs) DLD40 RoentDek. The three components of the velocity in (VH+,Ve) for each coincident DPI event are deduced from the impact positions of the particles at the PSDs and their TOF. The space focusing induced by the electrostatic lenses reduces the influence of the finite dimensions of the interaction region on the spatial resolution, while the global bending of the ion and/or electron trajectories preserving the time-of-flight information enables us to achieve an efficient collection of the emitted particles for a reduced magnitude of the extraction field. This property minimizes the influence of the finite temporal resolution, of the order of 150 ps in these experiments, mostly influencing the resolution of the electron versus velocity component parallel to the extraction field.
The instrumental widths are implemented in a Monte Carlo simulation of the charged particle trajectories which enables us to convolute the theoretical results with the apparatus function as presented below. Since the ion-fragment energy resolution is estimated at about 0.5 eV for extraction fields of the order of 100 V/cm, the evolution of the MFPADs as a function of
the proton kinetic energy (KE) will be presented for selection of the coincident events in bands of corresponding KE width.
The VC method allows for the construction of ion-electron kinetic-energy correlation diagrams (KECDs), which corre- sponds to a two-dimensional (2D) histogram of the (H+, e−) coincident events, deduced from the analysis of the moduli of the (VH+,Ve) vectors. This probability distribution is repre- sented as a function of the electron energy and the photoion ki- netic energy. A detailed analysis of the events in these KECDs for H2and D2DPI has been discussed elsewhere [44,46]. By integrating the KECDs over the electron energy one obtains the DPI cross section as a function of the photoion KE.
The FLN functions in Eq. (1) for each process appearing in the KECD diagrams with energies (EH+, Ee−) are extracted from all the coincident events (H+, e−) recorded with a 4π col- lection by the VC method, by performing a Legendre-Fourier analysis (with a three-angle fit) of the directly measured MFPAD I (χ ,θe,φe) distribution. Therefore MFPADs can be discriminated for any photoion KE or electron energy [32].
VI. RESULTS
We present results for MFPADs, CDAD, and total CD in DPI of both H2and D2 molecules, excited by three different photon energies, which represent distinct physical situations in terms of the role that autoionizing processes play in the photodynamics: 20 eV for H2 (and 19 eV for D2), 27 eV, and 32.5 eV. Figures2and3show the dissociative ionization probability components, which correspond (up to a scale factor 2π ) to the (θe,φe)-integrated quantity F00+ 0.25F20; the insets represent polar plots of the MFPADs resulting from Eq. (2) or Eq. (19) for orientational molecular geometry χ= 90◦ and φe= 90◦(left side) and φe= 270◦(right side), as a function of H+(D+) KE. At photon energies around 20 eV, the dominant process is direct nonresonant dissociative photoionization into the dissociative part of the H2+ 2g+(1sσg) channel (see Fig.1) for both final symmetries 1+u and 1 u (in the following, symbols within parentheses indicate the ionization channel nlλg,uand the symmetry of the final electronic state2S+1±g,u):
H2+ ω → H + H++ e−[1sσg,1u+(1 u)]. (22) This process is relevant only at very low H+(D+) kinetic energies, which is a consequence of the rapid decrease of the Franck-Condon overlap between the initial vibrational state of the H2 (D2) and the vibrational states corresponding to the dissociative continuum of H2+(D2+). At this photon energy, when linear polarization is used, the photoelectron emission is preferentially produced in the form of a p wave, indicating that the first partial wave for the ejected electron [σu(a=1)(πu(b=1)) for 1u+(1 u) final symmetries, respectively] dominates [27]. Thus, according to Eqs. (20) and (21), the dominant contributions to the MFPADs come from Le= 0, 1, and 2. When using left circularly polarized light with the polarization vector rotating in the Y Z plane, contributions from the molecular orientations parallel (1u+) and perpendicular (1 u) to the polarization axis are added coherently. Available experimental results in D2 for the DPI probability and the MFPADs are in very good agreement with the calculations.
0 1 2 3 4 5 6 7 8
H+ Kinetic Energy [eV]
0 1 2 3
Probability per eV
hν = 32.5 eV
1.75 eV < EH < 2.25 eV 4.75 eV < E
H < 5.25 eV 6.5 eV < E
H < 7 eV
H+
H x
2pσu 1sσg
(c)
0 1 2 3 4 5
H+ Kinetic Energy [eV]
0 1 2
Probability per eV
hν = 27 eV
1.75 eV < EH < 2.25 eV
3.25 eV < EH < 3.75 eV
H+
H x
(b)
0 0.5 1 1.5 2
H+ Kinetic Energy [eV]
0 2 4 6 8 10 12
Probability per eV
hν = 20 eV
EH = 0.49 eV
H+
H x
(a)
×10-5
×10-5
×10-5
FIG. 2. (Color online) Dissociative ionization probability in H2
as a function of the proton kinetic energy for left-handed circularly po- larized light for three photon energies: (a) ω= 20 eV, (b) ω = 27 eV, and (c) ω= 32.5 eV. The probability corresponds to the geometry displayed in Fig. 1 and it is equal to the quantity F00+ 0.25F20
after integrating Eq. (2) over θe and φe. Experiment and theory are normalized on the θe-integrated F00 summed in a broad KE range chosen for each photon energy. Red circles, experimental results;
black solid line, theoretical results; green solid line, theoretical results convoluted with the instrumental resolution; black dashed line, dissociative ionization probability associated with the 1sσgionization channel; black dash-dotted line, probability associated with the 2pσu
channel. Insets: Polar plots of the MFPADs in the Y Z plane at fixed proton kinetic energies averaged within the energy intervals indicated in the figure. Same notation as for the probabilities plus a blue dashed line indicating a fitting to the experimental results.
For a photon energy of 27 eV the autoionization process (AI) throughQ1doubly excited states is now an open channel (a process indicated qualitatively in Fig.1), i.e.,
H2+ ω → H∗∗2 (Q11u+)→ H + H++ e−(1sσg,1+u), (23)
